Question

The sum of the deviations of the variates from the arithmetic mean is always
(A) +1
(B) 0
(C) -1
(D) None of the above

Answer 1

Hint: First of all, take the numbers 3, 4, and 5. Now, calculate the arithmetic mean of the given data using the formula, \[\text {Arithmetic Mean}=\dfrac{\text {sum of all data}}{\text{total no of data}}\] . Then, get the value of the deviation of 3, 4, and 5 from the arithmetic mean. At last, calculate the sum of the deviations of the variates from the arithmetic mean.

Complete step by step answer:
According to the question, we are asked to find the sum of the deviations of the variates from the arithmetic mean.
First of all, let us take the numbers 3, 4, and 5 as our data.
Now, our data are 3, 4, and 5 …………………………………..(1)
We know the formula, \[\text {Arithmetic Mean}=\dfrac{\text {sum of all data}}{\text{total no of data}}\]…………………………………(2)
From equation (1), we have all the data.
The sum of all data = \[3+4+5=12\] …………………………………………..(3)
The total number of data = 3 ………………………………………….(4)
Now, from equation (2), equation (3), and equation (4), we get
The arithmetic mean of all data = \[\dfrac{12}{3}=4\] …………………………………….(5)
The deviation of 3 from 4 = \[3-4=-1\] ………………………………………(6)
The deviation of 4 from 4 = \[4-4=0\] ………………………………………(7)
The deviation of 5 from 4 = \[5-4=1\] ………………………………………(8)
From equation (6), equation (7), and equation (8), we have the values of the deviations.
Now, the sum of deviations of the variates from the arithmetic mean = \[-1+0+1=0\] .
Therefore, the sum of deviations of the variates from the arithmetic mean is zero.

So, the correct answer is “Option B”.

Note: Instead of solving by taking the numbers, we can also solve this question by using a property. We know the property that the sum of deviations of the variates from the arithmetic mean is zero. Therefore, the correct option is (B).